CombinatorialCase102
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Jump to navigationJump to searchThis case produces Partitions of the set {⍳M} into exactly N parts. As such, it produces a subset of 101, limiting the result to just those rows with M subsets.
 M labeled balls (1), N unlabeled boxes (0), at least one ball per box (2)
 Sensitive to ⎕IO
 Counted result is an integer scalar
 Generated result is a nested vector of nested integer vectors.
The count for this function is M SN2 N where M SN2 N calculates the Stirling numbers of the 2^{nd} kind.
For example:
If we have 4 labeled balls (❶❷❸❹) and 2 unlabeled boxes with at least one ball per box, there are 7 (↔ 4 SN2 2) ways to meet these criteria:







The diagram above corresponds to the nested array
⍪102 1‼4 2 1 2 3 4 1 2 4 3 1 2 3 4 1 3 4 2 1 3 2 4 1 4 2 3 1 2 3 4 ⍝ Partitions of {⍳M} into N parts ⍝ Labeled balls, unlabeled boxes, ≥1 # Balls per Box ⍝ The number to the right in parens ⍝ represent the corresponding row from ⍝ the table in case 101. ⍪102 1‼4 4 1 2 3 4 (15) ⍪102 1‼4 3 1 2 3 4 (5) 1 3 2 4 (8) 1 2 3 4 (11) 1 4 2 3 (12) 1 2 4 3 (13) 1 2 3 4 (14) ⍪102 1‼4 2 1 2 3 4 (2) 1 2 4 3 (3) 1 2 3 4 (4) 1 3 4 2 (6) 1 3 2 4 (7) 1 4 2 3 (9) 1 2 3 4 (10) ⍪102 1‼4 1 1 2 3 4 (1) ⍪102 1‼4 0
In general, this case is related to 101 through the following identities (after sorting the items):
101 1‼M N ↔ ⊃,/102 1‼¨M,¨0..N 102 1‼M N ↔ R {(⍺=≢¨⍵)/⍵} 101 1‼M N
and is related to 112 through the following identities:
102 1‼M N ↔ {(2≢/¯1,(⊂¨⍋¨⍵)⌷¨⍵)/⍵} 112 1‼M N a←⊃102 1‼M N b← 110 1‼M N 112 1‼M N ↔ ,⊂[⎕IO+2] a[;b]